Catalan's Conjecture

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Theorem

The only solution to the Diophantine equation:

$x^a - y^b = 1$

for $a, b > 1$ and $x, y > 0$, is:

$x = 3, a = 2, y = 2, b = 3$


Proof


Also known as

This result is also known as Mihăilescu's Theorem, for Preda V. Mihăilescu.


Also see


Source of Name

This entry was named for Eugène Charles Catalan.


Historical Note

Catalan's Conjecture was first put forward by Eugène Charles Catalan in $1844$.

It was proven in $2002$ by Preda V. Mihăilescu.


Sources